Rational Numbers | IT

Question 27

The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.

Solution

A taxicab number is a number that can be written as the sum of two positive cubes in two different ways.

Step 1 — List of Cubes

We need to find numbers that are sums of cubes. Let us list some perfect cubes. This will help us check our calculations. $1^3 = 1$ $2^3 = 8$ $3^3 = 27$ $4^3 = 64$ $5^3 = 125$ $6^3 = 216$ $7^3 = 343$ $8^3 = 512$ $9^3 = 729$ $10^3 = 1000$ $11^3 = 1331$ $12^3 = 1728$ $13^3 = 2197$ $14^3 = 2744$ $15^3 = 3375$ $16^3 = 4096$ $17^3 = 4913$ $18^3 = 5832$ $19^3 = 6859$ $20^3 = 8000$ $21^3 = 9261$ $22^3 = 10648$ $23^3 = 12167$ $24^3 = 13824$ $25^3 = 15625$

Step 2 — For 4104

We need to find two positive integers, let's call them $a$ and $b$ . Their cubes must add up to 4104. So, we are looking for $a^3 + b^3 = 4104$ . Let us assume $a \le b$ to find unique pairs. This means $a^3$ must be less than or equal to $4104 / 2 = 2052$ . From our list, $12^3 = 1728$ and $13^3 = 2197$ . So, $a$ can be any integer from 1 to 12.

Let us try values for $a$ : If $a=1$ , $1^3 = 1$ . We need $b^3 = 4104 - 1 = 4103$ . $4103$ is not a perfect cube from our list.

If $a=2$ , $2^3 = 8$ . We need $b^3 = 4104 - 8 = 4096$ . From our list, $4096 = 16^3$ . So, one way is $2^3 + 16^3$ . $2^3 + 16^3 = 8 + 4096$ $= 4104$

$\boxed{4104 = 2^3 + 16^3}$

Let us continue to find the second way: If $a=9$ , $9^3 = 729$ . We need $b^3 = 4104 - 729 = 3375$ . From our list, $3375 = 15^3$ . So, another way is $9^3 + 15^3$ . $9^3 + 15^3 = 729 + 3375$ $= 4104$

$\boxed{4104 = 9^3 + 15^3}$

We have found two ways for 4104.

Step 3 — For 13832

We need to find two positive integers, $a$ and $b$ . Their cubes must add up to 13832. So, we are looking for $a^3 + b^3 = 13832$ . Again, let us assume $a \le b$ . This means $a^3$ must be less than or equal to $13832 / 2 = 6916$ . From our list, $18^3 = 5832$ and $19^3 = 6859$ . Also, $20^3 = 8000$ , which is too large. So, $a$ can be any integer from 1 to 19.

Let us try values for $a$ : If $a=1$ , $1^3 = 1$ . We need $b^3 = 13832 - 1 = 13831$ . Not a perfect cube.

If $a=2$ , $2^3 = 8$ . We need $b^3 = 13832 - 8 = 13824$ . From our list, $13824 = 24^3$ . So, one way is $2^3 + 24^3$ . $2^3 + 24^3 = 8 + 13824$ $= 13832$

$\boxed{13832 = 2^3 + 24^3}$

Let us continue to find the second way: We will skip some intermediate values of $a$ for brevity, as the process is the same. If $a=18$ , $18^3 = 5832$ . We need $b^3 = 13832 - 5832 = 8000$ . From our list, $8000 = 20^3$ . So, another way is $18^3 + 20^3$ . $18^3 + 20^3 = 5832 + 8000$ $= 13832$

$\boxed{13832 = 18^3 + 20^3}$

We have found two ways for 13832.

Answer

(i) For 4104: $2^3 + 16^3$ and $9^3 + 15^3$ (ii) For 13832: $2^3 + 24^3$ and $18^3 + 20^3$

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Q27

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Q28

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